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Gödel began his 1951 Gibbs Lecture by stating: “Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics.” (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel’s first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arithmetic, there exists a sentence GF of the language of the system which is true but unprovable in that system. Gödel’s second incompleteness theorem states that no consistent formal system can prove its own consistency.

We prove that any 1-closed (see def 1.1) model of the Π 2 consequences of PA satisfies ¬Cons PA which gives a proof of the second Godel incompleteness theorem without the use of the Godel diagonal lemma. We prove a few other theorems by the same method.

Gödel's first incompleteness theorem shows that no axiomatic theory can prove all mathematical truths, while Gödel's second incompleteness theorem shows that a specific mathematical result is unprovable. A famous mathematician of the time, David Hilbert, had asked for a proof that an important axiomatic theory was consistent, and Godel showed that such a proof could not be carried out within the axiomatic theory itself, and presumably could therefore not be established in a convincing way outside of the theory either.

This thesis explores the significance of Godel's Theorem for an understanding of law as rules, and of legal adjudication as rule-following. It argues that Godel's Theorem, read through Wittgenstein's understanding of rules and language as a contextual activity, and through Derrida's account of 'undecidability,' offers an alternative account of the relationship of judging to justice. Instead of providing support for the 'indeterminacy' claim, Godel's Theorem illuminates the predicament of undecidability that structures any interpretation and every legal decision, and which constitutes the opening to justice. The first argument in this thesis examines Godel's proof, Wittgenstein's views on rules, and Derrida's undecidability, as manifestations of a common concern with the limits of what can be formalized. The meta-argument examines their misinterpretation and misappropriation within legal theory as a case study of just what they mean about meaning, context, and justice as necessarily co-implicated.

The Gödelian symphony -- Foundations and paradoxes -- This sentence is false -- The liar and Gödel -- Language and metalanguage -- The axiomatic method or how to get the non-obvious out of the obvious -- Peano's axioms -- And the unsatisfied logicists, Frege and Russell -- Bits of set theory -- The abstraction principle -- Bytes of set theory -- Properties, relations, functions, that is, sets again -- Calculating, computing, enumerating, that is, the notion of algorithm -- Taking numbers as sets of sets -- It's raining paradoxes -- Cantor's diagonal argument -- Self-reference and paradoxes -- Hilbert -- Strings of symbols -- In mathematics there is no ignorabimus -- Gödel on stage -- Our first encounter with the incompleteness theorem -- And some provisos -- Gödelization, or say it with numbers! -- TNT -- The arithmetical axioms of tnt and the standard model N -- The fundamental property of formal systems -- The Gödel numbering -- And the arithmetization of syntax -- Bits of recursive arithmetic -- Making algorithms precise -- Bits of recursion theory -- Church's thesis -- The recursiveness of predicates, sets, properties, and relations -- And how it is represented in typographical number theory -- Introspection and representation -- The representability of properties, relations, and functions -- And the Gödelian loop -- I am not provable -- Proof pairs -- The property of being a theorem of TNI (is not recursive!) -- Arithmetizing substitution -- How can a TNT sentence refer to itself? -- Fixed point -- Consistency and omega-consistency -- Proving G1 -- Rosser's proof -- The unprovability of consistency and the immediate consequences of G1 and -- G2 -- Technical interlude -- Immediate consequences of G1 and G2 -- Undecidable1 and undecidable 2 -- Essential incompleteness, or the syndicate of mathematicians -- Robinson arithmetic -- How general are Gödel's results? -- Bits of turing machine -- G1 and G2 in general -- Unexpected fish in the formal net -- Supernatural numbers -- The culpability of the induction scheme -- Bits of truth (not too much of it, though) -- The world after Gödel -- Bourgeois mathematicians! : the postmodern interpretations -- What is postmodernism? -- From Gödel to Lenin -- Is biblical proof decidable? -- Speaking of the totality -- Bourgeois teachers! -- (un)interesting bifurcations -- A footnote to Plato -- Explorers in the realm of numbers -- The essence of a life -- The philosophical prejudices of our times -- From Gödel to Tarski -- Human, too human -- Mathematical faith -- I'm not crazy! -- Qualified doubts -- From gentzen to the dialectica interpretation -- Mathematicians are people of faith -- Mind versus computer : Gödel and artificial intelligence -- Is mind (just) a program? -- Seeing the truth and going outside the system -- The basic mistake -- In the haze of the transfinite -- Know thyself : Socrates and the inexhaustibility of mathematics -- Gödel versus wittgenstein and the paraconsistent interpretation -- W

Mark Steiner criticizes some remarks Wittgenstein makes about Gödel. Steiner takes Wittgenstein to be disputing a mathematical result. The paper argues that Wittgenstein does no such thing. The contrast between the realist and the demonstrativist concerning mathematical truth is examined. Wittgenstein is held to side with neither camp. Rather, his point is that a realist argument is inconclusive.

In "Remarks on the Foundations of Mathematics" Wittgenstein discusses an argument that goes from Gödel’s incompleteness result to the conclusion that some truths of mathematics are unprovable. Wittgenstein takes issue with this argument. Wittgenstein’s remarks in this connection have received very negative reaction from some very prominent people, for example, Gödel and Dummett. The paper is a defense of what Wittgenstein has to say about the argument in question.

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.

Remarks on the Foundations of Mathematics, Wittgenstein, despite his official 'mathematical nonrevisionism', slips into attempting to refute Gödel's theorem. Actually, Wittgenstein could have used Gödel's theorem to good effect, to support his view that proof, and even truth, are 'family resemblance' concepts. The reason that Wittgenstein did not see all this is that Gödel's theorem had become an icon of mathematical realism, and he was blinded by his own ideology. The essay is a reply to Juliet Floyd's work on Gödel: what she says Wittgenstein said, I say he should have said, but didn't (couldn't).

O presente artigo procede, em primeiro lugar, a um exame das evidências disponíveis referentes à atitude de Wittgenstein em relação ao, bem como conhecimento do, primeiro teorema da incompletude de Gödel, incluindo as suas discussões com Turing, Watson e outros em 1937-1939, e o testemunho posterior de Goodstein e Kreisel Em segundo lugar, o artigo discute a importância filosófica e histórica da atitude de Wittgenstein em relação ao teorema de Gödel e outros teoremas da lógica matemática, contrastando esta atitude com a de, por exemplo, Penrose. Finalmente, a autora responde também a criticas instrutivas feitas por Mark Steiner a um artigo seu publicado em 1995, as quais estabelecem a importância do trabalho semântico de Tarski, quer para o nosso entendimento das observacoes de Wittgenstein. /// This paper presents, first, a survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937-1939, and later testimony of Goodstein and Kreisel Secondly, the article discusses the philosophical and historical importance of Wittgenstein's attitude toward Gödal's and other theorems in mathematical logic, contrasting this attitude with that of, e. g., Penrose. Finally, the author also replies to an instructive criticism of her 1995 paper by Mark Steiner which assesses the importance ofTarskih semantical work, both for our understanding of Wittgenstein's remarks on Godel, and our understanding ofGodeVs theorem itself.